3,941 research outputs found
Weighted Energy-Dissipation principle for gradient flows in metric spaces
This paper develops the so-called Weighted Energy-Dissipation (WED)
variational approach for the analysis of gradient flows in metric spaces. This
focuses on the minimization of the parameter-dependent global-in-time
functional of trajectories \mathcal{I}_\varepsilon[u] = \int_0^{\infty}
e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varepsilon}\phi(u(t))
\right) \dd t, featuring the weighted sum of energetic and dissipative
terms. As the parameter is sent to~, the minimizers
of such functionals converge, up to subsequences, to curves of
maximal slope driven by the functional . This delivers a new and general
variational approximation procedure, hence a new existence proof, for metric
gradient flows. In addition, it provides a novel perspective towards
relaxation
A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity
In this paper we discuss a family of viscous Cahn-Hilliard equations with a
non-smooth viscosity term. This system may be viewed as an approximation of a
"forward-backward" parabolic equation. The resulting problem is highly
nonlinear, coupling in the same equation two nonlinearities with the diffusion
term. In particular, we prove existence of solutions for the related initial
and boundary value problem. Under suitable assumptions, we also state
uniqueness and continuous dependence on data.Comment: Key words and phrases: diffusion of species; Cahn-Hilliard equations;
viscosity; non-smooth regularization; nonlinearities; initial-boundary value
problem; existence of solutions; continuous dependenc
Global attractors for gradient flows in metric spaces
We develop the long-time analysis for gradient flow equations in metric
spaces. In particular, we consider two notions of solutions for metric gradient
flows, namely energy and generalized solutions. While the former concept
coincides with the notion of curves of maximal slope, we introduce the latter
to include limits of time-incremental approximations constructed via the
Minimizing Movements approach.
For both notions of solutions we prove the existence of the global attractor.
Since the evolutionary problems we consider may lack uniqueness, we rely on the
theory of generalized semiflows introduced by J.M. Ball. The notions of
generalized and energy solutions are quite flexible and can be used to address
gradient flows in a variety of contexts, ranging from Banach spaces to
Wasserstein spaces of probability measures.
We present applications of our abstract results by proving the existence of
the global attractor for the energy solutions both of abstract doubly nonlinear
evolution equations in reflexive Banach spaces, and of a class of evolution
equations in Wasserstein spaces, as well as for the generalized solutions of
some phase-change evolutions driven by mean curvature
The catalytic role of beta effect in barotropization processes
The vertical structure of freely evolving, continuously stratified,
quasi-geostrophic flow is investigated. We predict the final state
organization, and in particular its vertical structure, using statistical
mechanics and these predictions are tested against numerical simulations. The
key role played by conservation laws in each layer, including the fine-grained
enstrophy, is discussed. In general, the conservation laws, and in particular
that enstrophy is conserved layer-wise, prevent complete barotropization, i.e.,
the tendency to reach the gravest vertical mode. The peculiar role of the
-effect, i.e. of the existence of planetary vorticity gradients, is
discussed. In particular, it is shown that increasing increases the
tendency toward barotropization through turbulent stirring. The effectiveness
of barotropisation may be partly parameterized using the Rhines scale . As this parameter decreases (beta increases) then
barotropization can progress further, because the beta term provides enstrophy
to each layer
Modeling and analysis of a phase field system for damage and phase separation processes in solids
In this work, we analytically investigate a multi-component system for
describing phase separation and damage processes in solids. The model consists
of a parabolic diffusion equation of fourth order for the concentration coupled
with an elliptic system with material dependent coefficients for the strain
tensor and a doubly nonlinear differential inclusion for the damage function.
The main aim of this paper is to show existence of weak solutions for the
introduced model, where, in contrast to existing damage models in the
literature, different elastic properties of damaged and undamaged material are
regarded. To prove existence of weak solutions for the introduced model, we
start with an approximation system. Then, by passing to the limit, existence
results of weak solutions for the proposed model are obtained via suitable
variational techniques.Comment: Keywords: Cahn-Hilliard system, phase separation, elliptic-parabolic
systems, doubly nonlinear differential inclusions, complete damage, existence
results, energetic solutions, weak solutions, linear elasticity,
rate-dependent system
Excitation and control of large amplitude standing magnetization waves
A robust approach to excitation and control of large amplitude standing
magnetization waves in an easy axis ferromagnetic by starting from a ground
state and passage through resonances with chirped frequency microwave or spin
torque drives is proposed. The formation of these waves involves two stages,
where in the first stage, a spatially uniform, precessing magnetization is
created via passage through a resonance followed by a self-phase-locking
(autoresonance) with a constant amplitude drive. In the second stage, the
passage trough an additional resonance with a spatial modulation of the driving
amplitude yields transformation of the uniform solution into a doubly
phase-locked standing wave, whose amplitude is controlled by the variation of
the driving frequency. The stability of this excitation process is analyzed
both numerically and via Whitham's averaged variational principle
- …